Hiking Austria - low budget, low equiptment, still possible?

Hey everyone,
I will be in Innsbruck, Austria for 3-5 days in early June and I am wondering if someone could help me plan the best way to see the Alps. Here is some relevant details.

- I have a lot of walking kilometers behind me but no experience above the snow line
- I have some rockclimbing experience but only as a second and not at high grades
- I have hiking boots, warm waterproof clothing etc but NO technical equipment (crampons, ropes etc)
- I have a tent, but its not a good one
- I would like to do some hiking at high altitude - glacier travel and/or climbing if possible
- I guess the best solution would be to get a guide or join an organised trek, but I have a very low budget :( .
- I would like to stay in one of the mountain huts (are they expensive???)

Thank you in advance for any ideas!